When multiplying two matrices, there's a manual procedure we all know how to go
through. Each result cell is computed separately as the dot-product of a row in
the first matrix with a column in the second matrix. While it's the easiest way
to compute the result manually, it may obscure a very interesting property of
the operation: multiplying A by B is the linear combination of A's columns
using coefficients from B. Another way to look at it is that it's a linear
combination of the rows of B using coefficients from A.

In this quick post I want to show a colorful visualization that will make this
easier to grasp.

Right-multiplication: combination of columns

Let's begin by looking at the right-multiplication of matrix X by a column
vector:

Representing the columns of X by colorful boxes will help visualize this:

Sticking the white box with a in it to a vector just means: multiply this
vector by the scalar a. The result is another column vector - a linear
combination of X's columns, with a, b, c as the coefficients.

Right-multiplying X by a matrix is more of the same. Each resulting column
is a different linear combination of X's columns:

Graphically:

If you look hard at the equation above and squint a bit, you can recognize this
column-combination property by examining each column of the result matrix.

Left-multiplication: combination of rows

Now let's examine left-multiplication. Left-multiplying a matrix X by a row
vector is a linear combination of X's rows:

Is represented graphically thus:

And left-multiplying by a matrix is the same thing repeated for every result
row: it becomes the linear combination of the rows of X, with the
coefficients taken from the rows of the matrix on the left. Here's the equation
form: